Properties

Label 2-5712-1.1-c1-0-2
Degree 22
Conductor 57125712
Sign 11
Analytic cond. 45.610545.6105
Root an. cond. 6.753556.75355
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 7-s + 9-s − 3·11-s + 5·13-s + 3·15-s − 17-s − 2·19-s + 21-s − 6·23-s + 4·25-s − 27-s − 6·29-s + 4·31-s + 3·33-s + 3·35-s + 11·37-s − 5·39-s − 12·41-s + 43-s − 3·45-s − 12·47-s + 49-s + 51-s − 9·53-s + 9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 0.774·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s − 1.25·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 0.507·35-s + 1.80·37-s − 0.800·39-s − 1.87·41-s + 0.152·43-s − 0.447·45-s − 1.75·47-s + 1/7·49-s + 0.140·51-s − 1.23·53-s + 1.21·55-s + ⋯

Functional equation

Λ(s)=(5712s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5712s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57125712    =    2437172^{4} \cdot 3 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 45.610545.6105
Root analytic conductor: 6.753556.75355
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5712, ( :1/2), 1)(2,\ 5712,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 0.52097696910.5209769691
L(12)L(\frac12) \approx 0.52097696910.5209769691
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
7 1+T 1 + T
17 1+T 1 + T
good5 1+3T+pT2 1 + 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 15T+pT2 1 - 5 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 14T+pT2 1 - 4 T + p T^{2}
37 111T+pT2 1 - 11 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1T+pT2 1 - T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 112T+pT2 1 - 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1T+pT2 1 - T + p T^{2}
83 115T+pT2 1 - 15 T + p T^{2}
89 19T+pT2 1 - 9 T + p T^{2}
97 1+19T+pT2 1 + 19 T + p T^{2}
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   L(s)=p j=12(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.019507455750696322517359396577, −7.60071241079856865939505009051, −6.52974126119963944803026329454, −6.15891201432989065467850720102, −5.20983774777158442191812465551, −4.35865858694941866044297529040, −3.78371489917505067880314892179, −3.02402476613205357995053379893, −1.73527382204318293609960924592, −0.38832682507999745116601502102, 0.38832682507999745116601502102, 1.73527382204318293609960924592, 3.02402476613205357995053379893, 3.78371489917505067880314892179, 4.35865858694941866044297529040, 5.20983774777158442191812465551, 6.15891201432989065467850720102, 6.52974126119963944803026329454, 7.60071241079856865939505009051, 8.019507455750696322517359396577

Graph of the ZZ-function along the critical line